3D cursor orientation
Activate the 3D Cursor in the Toolbar T.
Open up the Sidebar (N) and, on the Tool tab, change the Orientation to Geometry.
Click on the surface of the object to update the orientation of the 3D Cursor.


If the 3D Cursor is not "sticking" to the surface, enable Cursor Surface Project in Preferences (Ctrl+Alt+U) > Editing > 3D Cursor.